Abstract
For practitioners of equity markets, option pricing is a major challenge during high volatility periods and Black-Scholes formula for option pricing is not the proper tool for very deep out-of-the-money options. The Black-Scholes pricing errors are larger in the deeper out-of-the money options relative to the near the-money options, and it's mispricing worsens with increased volatility. Experts opinion is that the Black-Scholes model is not the proper pricing tool in high volatility situations especially for very deep out-of-the-money options. They also argue that prior to the 1987 crash, volatilities were symmetric around zero moneyness, with in-the-money and out-of-the money having higher implied volatilities than at-the-money options. However, after the crash, the call option implied volatilities were decreasing monotonically as the call went deeper into out-of-the-money, while the put option implied volatilities were decreasing monotonically as the put went deeper into in-the-money. Since these findings cannot be explained by the Black-Scholes model and its variations, researchers searched for improved option pricing models. Feedforward networks provide more accurate pricing estimates for the deeper out-of-the money options and handles pricing during high volatility with considerably lower errors for out-of-the-money call and put options. This could be invaluable information for practitioners as option pricing is a major challenge during high volatility periods. In this article a nonparametric method for estimating S&P 100 index option prices using artificial neural networks is presented. To show the value of artificial neural network pricing formulas, Black-Scholes option prices are compared with the network prices against market prices. To illustrate the practical relevance of the network pricing approach, it is applied to the pricing of S&P 100 index options from April 4, 2014 to April 9, 2014. On the five days data while Black-Scholes formula prices have a mean $10.17 error for puts, and $1.98 for calls, while neural network’s error is less than $5 for puts, and $1 for calls.
Highlights
Much of the success and growth of the market for options and derivative securities may be traced to the much quoted articles by (Black-Scholes 1973) and (Merton, 1973), in which closed-form option pricing formulas were obtained through a dynamic hedging argument and arbitrage freeness condition
Variables x = S, r, s, T are entered into artificial neural networks in two different versions of the volatilitys. a) Implied volatility computed from Black-Scholes formula, b) Implied volatility computed by the use of ANN’s (Can, and Fadda, 2014), c) Historical volatility
Figure 5. and Table 2. depicts the relationship between the prices computed by artificial neural networks trained with implied volatility, and market prices
Summary
Much of the success and growth of the market for options and derivative securities may be traced to the much quoted articles by (Black-Scholes 1973) and (Merton, 1973), in which closed-form option pricing formulas were obtained through a dynamic hedging argument and arbitrage freeness condition. On the other hand Macbeth and Merville (Merville, 1979) revealed that the calculated with the implied volatility, Black-Scholes prices of at-or nearthe-money options, are on average less than market prices for in-the-money call options. The extent to which it overprices an out-of-the-money option increases with the extent to which the option is out-of-the-money and decreases as time-to-maturity decreases It means that the implied volatilities are inversely related to the exercise price, and this fact is contrary to Black’s (Black, 1976) results. Cao and Chen (Bakshi, et al, 1997) provide closed form solutions for valuing options under stochastic volatility and stochastic interest rates using Heston’s (Heston, 1993) Fourier inversion method to calculate volatility and interest rate market risk premiums Their results document that stochastic volatility and stochastic interest rate models are structurally misspecified.
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